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title: Interactive HTML Slides
for Geometry Processing subtitle: using the PMP library author: Mario Botsch affiliation: Graphics & Geometry Group
Bielefeld University ...

How to use these HTML slides

  • Use the cursor keys left/right to navigate through the slides
  • Click page number (bottom right) to open navigation menu
  • Press f/ESC to enter/leave fullscreen mode
  • Press o or ESC to enter/leave overview mode
  • Double-click an item (e.g. an image) to zoom in/out.

Interactive Demo Applications

  • The demos are written in C++ using the PMP library
  • They are cross-compiled to Javascript using emscripten
  • They require support for WebGL 2
    • See https://caniuse.com/webgl2
    • The demos work nicely on Chrome, Chromium, or Firefox
    • They do not run on Apple Safari in MacOS and iOS

Subdivision Surfaces {.section}

Loop Subdivision

  • Subdivision scheme for triangle meshes
  • Generates $C^2$ continuous limit surfaces:
    • $C^1$ for extraordinary vertices (valence ≠ 6)
    • $C^2$ continuous everywhere else

Loop, Smooth Subdivision Surfaces Based on Triangles, M.S. thesis, 1987{target="_blank" .footer}

Loop Subdivision Rules

New edge vertices HSPACE(100px) Update old vertices (valence $k$)

Loop, Smooth Subdivision Surfaces Based on Triangles, M.S. thesis, 1987{target="_blank" .footer}

Generalized Catmull-Clark Subdivision

  • Subdivision scheme for arbitrary polygons
  • Connect new face points to edge-vertex-edge triple
    • Turns all polygon faces into quads
  • Generates $C^2$ continuous limit surfaces:
    • $C^1$ for extraordinary vertices (valence ≠ 4)
    • $C^2$ continuous everywhere else

DeRose et al, Subdivision Surfaces in Character Animation, SIGGRAPH 1998{target="_blank" .footer}

Catmull-Clark Rules

VSPACE(50px)

::: col30 New face vertices

::: tiny $$\vec{f} = \frac{1}{n} \sum_{i=1}^n \vec{v}_i$$ ::: :::

::: col30 New edge vertices

::: tiny $$\vec{e} = \frac{1}{4} \left(\vec{v}_1 + \vec{v}_2 + \vec{f}_1 + \vec{f}_2\right)$$ ::: :::

::: col40 Update old vertices (valence $k$)

::: tiny $$ \vec{v} = \frac{k-2}{k} \vec{v}

  • \frac{1}{k^2} \sum_{i=1}^k \vec{v}_i
  • \frac{1}{k^2} \sum_{i=1}^k \vec{f}_i $$ ::: :::

DeRose et al, Subdivision Surfaces in Character Animation, SIGGRAPH 1998{target="_blank" .footer}

Try it yourself!

pure quad mesh (triangulate for Loop subdivision){ #demo width=1000px height=550px }

Try it yourself!

mixed tri-quad mesh{ width=1000px height=550px }

[Model created using Blender, original from Willem-Paul van Overbruggen]{.footer}

Surface Smoothing {.section}

Diffusion Flow on Meshes

  • Continuous PDE: $\frac{\partial \vec{x}}{\partial t} ;=; \lambda \Delta \vec{x}$
  • Explicit integration per vertex: $\vec{x}_i \leftarrow \vec{x}_i + \delta t , \lambda \Delta \vec{x}_i$

0 iterations{ height=300px } 10 iterations{ height=300px } 100 iterations{ height=300px }

Uniform Laplace Discretization

$$ \laplace \vec{x}\of{v_i} ;:=; \frac{1}{\abs{\set{N}1\of{v_i}}} \sum{v_j \in \set{N}_1\of{v_i}} \left( \vec{x}\of{v_j} - \vec{x}\of{v_i} \right) $$

7 3

  • Properties
    • simple and efficient
    • depends only on connectivity
    • does not take into account geometry at all

{ height=200px }

Taubin, A Signal Processing Approach to Fair Surface Design, SIGGRAPH 1995{.footer target="_blank"}

Cotan Laplace Discretization

$$ \laplace \vec{x}\of{v_i} ;:=; \frac{1}{2A\of{v_i}} \sum_{v_j \in \set{N}1\of{v_i}} \left( \cot \alpha{ij} + \cot \beta_{ij} \right) \left( \vec{x}\of{v_j} - \vec{x}\of{v_i} \right) $$

7 3

  • Properties
    • takes geometry and connectivity into account
    • more accurate discretization
    • can be derived through FEM

{ height=200px }

Meyer et al, Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, VisMath III, 2003{target="_blank" .footer}

Uniform or Cotan Discretization?

5 5

  • Uniform Laplacian is an inaccurate discretization
  • Might be non-zero even for planar meshes
  • Smoothes geometry and triangulation
  • Might be desired for mesh regularization

{ width=400px }

Desbrun et al, Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow, SIGGRAPH 1999{target="_blank" .footer}

Numerical Integration

  • Let's write the position update in matrix notation
  • Write all points $\vec{x}_i^{(t)}$ in a large vector/matrix: $$\vec{X}^{(t)} = \trans{\left( \vec{x}_1^{(t)}, \ldots, \vec{x}_n^{(t)} \right)} \in \R^{n\times 3}$$
  • Matrix version of explicit integration $$\vec{X}^{(t+1)} = (\vec{I} + \delta t , \lambda \vec{L}) , \vec{X}^{(t)}$$
  • Matrix version of implicit integration $$(\vec{I} - \delta t , \lambda \vec{L}) , \vec{X}^{(t+1)} = \vec{X}^{(t)}$$

HINT({{Easy to implement, but requires small $\delta t \lambda$ for stability}}, 750, 400) HINT({{Works for any $\delta t \lambda$, but has to solve linear system(s)}}, 750, 530)

Desbrun et al, Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow, SIGGRAPH 1999{target="_blank" .footer}

Try it yourself!

{ #demo width=1000px height=600px }

Isotropic Remeshing {.section}

Isotropic Triangle Remeshing

1 1 1

original mesh{ height=500px }

uniform remeshing{ height=500px }

adaptive remeshing{ height=500px }

Uniform Remeshing

VSPACE(50px)

  • Specify target edge length $L$
  • Iterate a few times
    1. Split edges longer than $\frac{4}{3} L$
    2. Collapse edges shorter than $\frac{4}{5}L$
    3. Flip edges to get closer to valence 6
    4. Shift vertices by tangential relaxation
    5. Project vertices onto input mesh

Botsch & Kobbelt, A Remeshing Approach to Multiresolution Modeling, SGP 2004{target="_blank" .footer}

Local Remeshing Operators

{ height=500px } { height=500px } { height=500px } { height=500px }

Botsch & Kobbelt, A Remeshing Approach to Multiresolution Modeling, SGP 2004{target="_blank" .footer}

Adaptive Remeshing

6 4

  • Adapt edge length to local curvature
  • Compute maximum principle curvature on reference mesh
  • Determine local target edge length from max-curvature
  • Adjust split & collapse criteria accordingly

{ width=400px }

Dunyach et al, Adaptive Remeshing for Real-Time Mesh Deformation, EG 2013{target="_blank" .footer}

Real-Time Remeshing

{ .autoplay .controls height=600px }

Dunyach et al, Adaptive Remeshing for Real-Time Mesh Deformation, EG 2013{target="_blank" .footer}

Real-Time Remeshing

{ .autoplay .controls height=600px }

Dunyach et al, Adaptive Remeshing for Real-Time Mesh Deformation, EG 2013{target="_blank" .footer}

Let's try!

{ #demo width=1000px height=600px }

Feature Preservation

6 4

  • Define feature edges / vertices
    • Large dihedral angles
    • Material boundaries
  • Adjust local operators
    • Don’t flip feature edges
    • Collapse only along features
    • Univariate smoothing
    • Project to feature curves
    • Don’t touch feature vertices

{ width=400px }

Let's try!

{ #demo width=1000px height=600px }